Solving exponential equations with fractional and variable exponents

[Cyrus200]

ok a friend told me that this is a great forum.
yeah i need two questions...ahem i feel helpless so bare with me...

Legend-To the power is (^)

Solve for X

1)8^1/4 times (1/4)^x over 2 = 16^3/4

2)5^24x^2 = 5^-71x+30

Im new here and i will spending a lot of time here because Math isn't one of my strongest subjects, I am in grd.11 advance math btw.

 

[Cyrus200]

Oh I am deeply sorry I posted in the wrong board...*sigh*
Would one of the mods. erase this...sorry again

 

[HallsofIvy]

Nothing wrong with feeling insignificant!

Also nothing wrong with posting mathematics questions to the mathematics board!

Although you might get a more sympathetic hearing for problems like these by posting to the "homework" boards.


In any case, you have:
1)8^1/4 times (1/4)^x over 2 = 16^3/4

By which I guess you mean 8^(1/4)*(1/4)^x/2= 16^(3/4)

First thing you should do is clear the "constant" parts from the left side by dividing the equation by 8^(1/4) and multiplying by 2. That gives (1/4)^x = 2(16^(3/4))/8^(1/4). It might help to break 16 into 2*8 and write 16^(3/4) as 2^(3/4)*8^(3/4). 8^(3/4)= (8^(1/4))^3 so 8^(3/4)/8^(1/4)= (8^(1/4))^2= 8^(1/2)= (4*2)^(1/2)= 2*2^(1/2).

That simplifies the right side to give (1/4)^x= 4*2^(1/2).

Since x is an exponent on the right, you need to use the "opposite" of exponential, the logarithm.
log((1/4)^x)= xlog(1/4)= log(4*2^(1/2)= log(4)+ (1/2)log(2)
so x= (log(4)+ (1/2)log(2))/(log(1/4))

Since log(4)= log(2^2)= 2log(2) and log(1/4)= -log(4)= -2log(2),

x= (2log(2)+ (1/2)log(2))/(-2log(2))
= (2+ 1/2)log(2)/(-2 log(2))= (5/2)/(-2)= -5/4.

2)5^24x^2 = 5^-71x+30

Here we really need more information- more parentheses!

Does 5^24x^2 mean 5^(24x^2)?. In other words is the x^2 in the exponent as well?

Also does 5^-71x+ 30 mean (5^(-71))x+ 30
or (5^(-71x))+ 30 or 5^(-71x+ 30)?

 

[H20]

Sky high! w00t! Go fish! MONG

 

[vsage]

couldn't hurt to use a few more parentheses. Please edit your post if you don't already have the solutions?

 

[baby_garfield]

the first one is kinda confusing cause the way you typed it. but the second one is not hard. since both sides have the base of 5, and they are equal, that means the exponents have to be the same. so 24x^2=-71x+30

 

[mathwonk]

this thread reminds me of homer and jethro: "lord, I feel so unnecessary"

 

[ksinclair13]

#1
If you meant
((8^.25)(.25^x))/2 = 16^.75
then, x = -1.625.